Asymptotics for Two-Dimensional Atoms

Abstract

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z > 0 and N quantum electrons of charge −1 is \({E(N,Z)=-\frac{1}{2}Z^2{\rm ln} Z+(E^{\rm TF}(\lambda)+\frac{1}{2}c^{\rm H})Z^2+o(Z^2)}\) when Z → ∞ and N/Z → λ, where ETF(λ) is given by a Thomas–Fermi type variational problem and cH ≈ −2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z → ∞, which is contrary to the expected behavior of three-dimensional atoms.